For any [itex]a_1,\cdots , a_N \geq 0[/itex] and [itex]p\geq 1[/itex]:

[itex] \sum_j (a_j)^p \geq (a_j)^p [/itex] for all [itex]j \in \mathbb{N}[/itex]. Taking the maximum over [itex]j[/itex] of both sides yields the lower bound.

Notice that [itex](a_j)^p \leq (\text{max}_ja_j)^p [/itex] for all [itex]j\in\mathbb{N}[/itex]. Summing both sides over [itex]j[/itex] yields the upper bound.

So these lead to the inequalities [tex] \sum_j (a_j)^p \geq (\text{max}_j a_j )^p[/tex] and [tex]\displaystyle \sum_j (a_j)^p \leq N( \text{max}_j a_j )^p[/tex] which leads to (a)(i) and (a)(ii) for [itex]p=1,2[/itex].